Electron. J. Diff. Equ.,
Vol. 2009(2009), No. 165, pp. 18.
An optimal existence theorem for positive solutions
of a fourpoint boundary value problem
Man Kam Kwong, James S. W. Wong
Abstract:
We are interested in the existence of positive solutions to a
fourpoint boundary value problem of the differential equation
on
.
The value of
at 0 and 1 are each a multiple of
at an interior point.
Many known existence criteria are based on the limiting values
of
as
approaches
and infinity.
In this article we obtain an optimal criterion (thereby improving
all existing results of kind mentioned above) by comparing these
limiting values to the smallest eigenvalue of the corresponding
fourpoint problem of the associated linear equation.
In the simpler case of threepoint boundary value problems, the
same result has been established in an earlier paper by the first
author using the shooting method.
The method of proof is based upon a variant of Krasnoselskii's fixed
point theorem on cones, the classical KreinRutman theorem,
and the Gelfand formula relating the spectral radius of a linear
operator to its norm.
Submitted February 12, 2009. Published December 22, 2009.
Math Subject Classifications: 34B10, 34B15, 34B18.
Key Words: Fourpoint boundary value problem; secondorder ODE;
Krasnoselskii fixed point theorem; mappings on cones.
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Man Kam Kwong
Department of Applied Mathematics
Hong Kong Polytechnic University,
Hong Kong, SAR, China
email: mankwong@polyu.edu.hk


James S. W. Wong
Institute of Mathematical Research
Department of Mathematics,
University of Hong Kong
Hong Kong, SAR, China
email: jsww@chinneyhonkwok.com

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